Claim . Let H = k{zij., ,,Zm}y y := Zm and / С ff be an ideal such that dy(I) Ç J, where dy := д/ду. Then / can be generated by functions from H not depending on y.

In our situation there is a simple direct argumentation. We may €tssume the ideal / to be generated by polynomials /i,..., /p* Let Ф* С k[ziy..., Zm-i] be the ideal of all polynomiab (p, such that <py^ is the leading term (with respect to y) of a polynomial in /. viously Фо С Ф1 С Ф2— But the assumption dy{I) С / impUes ф^ = Ф1 = Ф2 = ..., hence / is generated by Ф^,.

Proof of the Proposition. Any nontrivial element v € Вег(А) is induced by some derivation V € Der(ff). Suppose that deg(v) < 0. Hence, V = J^aid/dzi, a^ € A;, and we may assume ai ф 0. After an appropriate linear change of coordinates the derivation V reads dfdy^ у £ H, Since, v € Der(A), we have д/ду{1) Ç /. It foUows from the Claim that the ideal / can be generated by elements of H not depending on y. Hence, dim A = dimff/J > 1 - which dicts ош" assumption.

Remark . In case A is the analytic algebra associated with a normal germ of positive dimension, one can prove ([Wa], (1.6)) the tion using the well-known Zariski's lemma. It seems that the present proof can be adapted to other (not only homogeneous) cases.

The purpose of the rest of this section is to prove some partial cases of the following conjecture (cf. also [A2], [A3]).

Conjecture . Let A be an artinian ^-|.-graded complete tion, i.e. all weights Wi, i = 1,... ,m, are positive. Then there are no derivations on A of negative weight: T^{A)- = Der(i4)-. = 0.

Example . Put V = z^xd/dx + z^ydjdy in the notations of the ample in Section 3 . It is easy to see that V defines the derivation t; of the artinian algebra A and deg(v) = — 2 < 0!

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